Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion

The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $\hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve which provides simple formulas for $\psi$-class intersections numbers and Hodge integrals on $\overline{\mathcal{M}}_{g,n}$.